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Boosting Graph Pooling with Persistent Homology

Chaolong Ying, Xinjian Zhao, Tianshu Yu

TL;DR

A novel mechanism for injecting global topological invariance into pooling layers using PH, motivated by the observation that filtration operation in PH naturally aligns graph pooling in a cut-off manner is investigated, leading to improved performance.

Abstract

Recently, there has been an emerging trend to integrate persistent homology (PH) into graph neural networks (GNNs) to enrich expressive power. However, naively plugging PH features into GNN layers always results in marginal improvement with low interpretability. In this paper, we investigate a novel mechanism for injecting global topological invariance into pooling layers using PH, motivated by the observation that filtration operation in PH naturally aligns graph pooling in a cut-off manner. In this fashion, message passing in the coarsened graph acts along persistent pooled topology, leading to improved performance. Experimentally, we apply our mechanism to a collection of graph pooling methods and observe consistent and substantial performance gain over several popular datasets, demonstrating its wide applicability and flexibility.

Boosting Graph Pooling with Persistent Homology

TL;DR

A novel mechanism for injecting global topological invariance into pooling layers using PH, motivated by the observation that filtration operation in PH naturally aligns graph pooling in a cut-off manner is investigated, leading to improved performance.

Abstract

Recently, there has been an emerging trend to integrate persistent homology (PH) into graph neural networks (GNNs) to enrich expressive power. However, naively plugging PH features into GNN layers always results in marginal improvement with low interpretability. In this paper, we investigate a novel mechanism for injecting global topological invariance into pooling layers using PH, motivated by the observation that filtration operation in PH naturally aligns graph pooling in a cut-off manner. In this fashion, message passing in the coarsened graph acts along persistent pooled topology, leading to improved performance. Experimentally, we apply our mechanism to a collection of graph pooling methods and observe consistent and substantial performance gain over several popular datasets, demonstrating its wide applicability and flexibility.
Paper Structure (36 sections, 5 theorems, 10 equations, 9 figures, 9 tables)

This paper contains 36 sections, 5 theorems, 10 equations, 9 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Graph Neural Networks.
  5. Dense Graph Pooling.
  6. Topological Features of Graphs.
  7. Methodology
  8. Overview
  9. Topology-Invariant Pooling
  10. Resampling.
  11. Persistence Injection.
  12. Topological Loss Function.
  13. Analysis
  14. Experiments
  15. Experimental Setup
  16. ...and 21 more sections

Key Result

Theorem 1

The self-loop augmented 1-dimensional topological features computed by PH is sufficient enough to be at least as expressive as 1-WL in terms of distinguishing non-isomorphic graphs with self-loops, i.e. if the 1-WL label sequences for two graphs $\mathcal{G}$ and $\mathcal{G}'$ diverge, there exists

Figures (9)

  • Figure 1: Illustration of Graph Pooling (GP) and Persistent Homology (PH). (a) GP and PH share a similar hierarchical fashion by coarsening a graph. (b) As a motivating experiment, we gradually change pooling ratio and count how persistence ratio (ratio of non-zero persistence) changes with it. (c) Illustration of persistence diagrams.
  • Figure 2: Overview of our method. The shaded part is one layer of Topology-Invariant Pooling.
  • Figure 3: Coarsened graphs from different methods in the preserving topological structure experiment.
  • Figure 4: Graphs pooled with different methods in graph classification experiment.
  • Figure 5: The training curves of DiffPool-TIP and DiffPool-TIP-NL on ENZYMES dataset. We show the average values and min-max range of objective and Wasserstein distance for multiple runs.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 0
  • Theorem 1
  • proof
  • Proposition 1
  • Lemma 1
  • proof